The curse of three dimensions: Why your brain is lying to you

Susan VanderPlas

Outline

Biology

Why evolution doesn’t prefer statisticians

The Sine Illusion

Three Dimensions? Or Two?

Statistical Graphics

Quantifying the brain-paper divide experimentally

Conclusions

Knowing is half the battle

Biology

Perception is three-dimensional

Perception is three-dimensional

Graphics are two-dimensional

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Perception is three-dimensional

Ambiguous Figures


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Artistic Trickery

The Sine Illusion

The Sine Illusion

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Three Dimensional Context

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Example: US Gas Prices


US Gas prices from 1995 to 2014 steadily increase over the time frame, with some dramatic short-term changes.

Source: EIA Weekly Retail Gasoline and Diesel Prices

Example: US Gas Prices


Standard deviation of daily gas prices between 1995 and 2014. The doubling of the standard deviation over the time frame is masked in the scatterplot of the data

Understanding the Illusion

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We can correct the illusion by increasing the blue line’s length until the red line is equal to the actual distance

How much is your brain lying to you?

Tufte’s Lie Factor


\[ \text{Lie Factor} = \frac{\text{Effect Size}_{graphic}}{\text{Effect Size}_{data}} \]



Psychological Lie Factor


\[ \text{Lie Factor (Brain)} = \frac{\text{Effect Size}_{perception}}{\text{Effect Size}_{graphic}} \]

Experiment Goals

  • Is the sine illusion a significant factor in reading simple plots?
    Do participants always choose the uncorrected plot with uniform line length?

  • How large is the effect of the sine illusion? Does it depend on the underlying function?

  • Would two individuals have similar psychological lie factors?

Experimental Design

  • Show participants sets of 6 plots with the sine illusion corrected to different degrees (including some overcorrection)
    • 3 underlying functions: sin(x), exp(x), and 1/x.

  • Participants choose the plot of the 6 shown which has lines that are the most uniform in length

  • Participants had to complete at least 10 trials to be included in the study

Experimental Design

Experimental Design

  • Compute the “Psychological Lie factor”, \(D^\ast\), for the chosen plot
    Uncorrected plot has constant line length of 1, so \(D\) is equivalent to the maximum line length in a sub plot

  • Normalize the chosen lie factor so that the lowest possible selection corresponds to 1. \[P = \frac{D^\ast}{D_{min}} \;\;\; 1\leq P \leq D^\ast\]
    Some test plots do not contain an uncorrected sub-plot, so we will work with \(P\) instead of \(D^\ast\) to maintain the scale of Tufte’s lie factor

Experimental Design

  • Compute the “Psychological Lie factor”, \(D^\ast\), for the chosen plot

  • Normalize the chosen lie factor so that the lowest possible selection corresponds to 1.


    106 participants completed 1542 trials
    (411 exponential, 316 inverse, and 815 sine)

Model Details


Hierarchical Bayesian model for \(\theta\), the overall lie factor

- Uniform priors on \(1\leq \mu\leq 4\), \(.1\leq\sigma\leq 2\)

- \(f(\theta | P, \mu, \sigma)\sim \text{Truncated Normal}(\mu, \sigma)\), \(\theta>1\)

Results

4 individuals who completed the most trials

Results

Overall Results

Results

Individual and Overall Credible Intervals

Conclusions

Conclusions

  • The sine illusion has a significant effect on our perception of simple plots

  • The effect does depend on the function

  • The effect seems to be fairly consistent across individuals, but there is some variance